On (1, $$\epsilon $$ϵ)-Restricted Max–Min Fair Allocation Problem

We study the max–min fair allocation problem in which a set of m indivisible items are to be distributed among n agents such that the minimum utility among all agents is maximized. In the restricted setting, the utility of each item j on agent i is either 0 or some non-negative weight $$w_j$$wj. For this setting, Asadpour et al. (ACM Trans Algorithms 8(3):24, 2012) showed that a certain configuration-LP can be used to estimate the optimal value to within a factor of $$4+\delta $$4+δ, for any $$\delta >0$$δ>0, which was recently extended by Annamalai et al. (in: Indyk (ed) Proceedings of the twenty-sixth annual ACMSIAM symposium on discrete algorithms, SODA 2015, San Diego, CA, USA, January 4–6, 2015) to give a polynomial-time 13-approximation algorithm for the problem. For hardness results, Bezáková and Dani (SIGecom Exch 5(3):11–18, 2005) showed that it is $$\mathsf {NP}$$NP-hard to approximate the problem within any ratio smaller than 2. In this paper we consider the $$(1,\epsilon )$$(1,ϵ)-restricted max–min fair allocation problem in which each item j is either heavy $$(w_j = 1)$$(wj=1) or light $$(w_j = \epsilon )$$(wj=ϵ), for some parameter $$\epsilon \in (0,1)$$ϵ∈(0,1). We show that the $$(1,\epsilon )$$(1,ϵ)-restricted case is also $$\mathsf {NP}$$NP-hard to approximate within any ratio smaller than 2. Using the configuration-LP, we are able to estimate the optimal value of the problem to within a factor of $$3+\delta $$3+δ, for any $$\delta >0$$δ>0. Extending this idea, we also obtain a quasi-polynomial time $$(3+4\epsilon )$$(3+4ϵ)-approximation algorithm and a polynomial time 9-approximation algorithm. Moreover, we show that as $$\epsilon $$ϵ tends to 0, the approximation ratio of our polynomial-time algorithm approaches $$3+2\sqrt{2}\approx 5.83$$3+22≈5.83.